Differentiation Rules

Constant Multiple and Sum Rules

Differentiate a polynomial using the constant multiple and sum rules.

constant-multiple-sum The sum rule: d/dx[f + g] = f' + g'. The constant multiple rule: d/dx[c·f] = c·f'. Apply both rules term by term to differentiate polynomials.

Example

Differentiate each term and preserve the constant multiples.

highlighted = computed this step

Step 1 — Set up

Start with the polynomial sum.

f(x)=3x4−2x2+5f(x)= 3x^{4}-2x^{2}+5

Step 2 — Differentiate first term

Multiply 3 by 4 and lower the exponent to 3.

3⋅4x3=12x3\hl{3} \cdot \hl{4} x^{ \hl{3} }= \hl{12} x^{ 3 }

Step 3 — Differentiate second term

Keep the minus sign and multiply 2 by 2 to get 4.

−2⋅2x=−4x- \hl{2} \cdot \hl{2} x=- \hl{4} x

Step 4 — Differentiate constant

The derivative of constant 5 is 0.

ddx5=0\frac{d}{dx} 5 = \hl{0}

Step 5 — Result

Combine the term derivatives.

f′(x)=12x3−4xf'(x)= \hlmath{12x^{3}-4x}